A014080 Factorions: equal to the sum of the factorials of their digits in base 10 (cf. A061602).

1, 2, 145, 40585

Offset

1,2

Comments

Poole (1971) showed that there are no further terms. - N. J. A. Sloane, Mar 17 2019

Base 6 also has four factorions, as does base 15. - Alonso del Arte, Oct 20 2012

This is row 10 of the table A193163. - M. F. Hasler, Nov 25 2015

References

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 145, p. 50, Ellipses, Paris 2008.

P. Kiss, A generalization of a problem in number theory, Math. Sem. Notes Kobe Univ., 5 (1977), no. 3, 313-317. MR 0472667 (57 #12362).

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see pp. 68, 305.

Joe Roberts, "The Lure of the Integers", page 35.

D. Wells, Curious and interesting numbers, Penguin Books, p. 125.

Links

Table of n, a(n) for n=1..4.

Project Euler, Problem 34: Digit factorials

P. Kiss, A generalization of a problem in number theory, [Hungarian], Mat. Lapok, 25 (No. 1-2, 1974), 145-149.

G. D. Poole, Integers and the sum of the factorials of their digits, Math. Mag., 44 (1971), 278-279, [JSTOR].

H. J. J. te Riele, Iteration of number-theoretic functions, Nieuw Archief v. Wiskunde, (4) 1 (1983), 345-360. See Example I.1.b.

Eric Weisstein's World of Mathematics, Factorion

Formula

If n has digits (d1,d2,...,dk) base 10, then n is on this list if and only if n = d1! + d2! + ... + dk!.

Example

1! + 4! + 5! = 1 + 24 + 120 = 145, so 145 is in the sequence.

Mathematica

Select[Range[50000], Plus @@ (IntegerDigits[ # ]!) == # &] (* Alonso del Arte, Jan 14 2008 *)

Prog

(J) (#~ (= +/@:!@:("."0)@":"0)) i.1e5 NB. Stephen Makdisi, May 14 2016
(Python)
from itertools import count, islice
def A014080_gen(): # generator of terms
    return (n for n in count(1) if sum((1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880)[int(d)] for d in str(n)) == n)
A014080_list = list(islice(A014080_gen(), 4)) # Chai Wah Wu, Feb 18 2022

Crossrefs

Cf. A061602, A193163, A214285, A254499, A306955.

Sequence in context: A254499 A071064 A265442 * A182478 A214910 A328228

Adjacent sequences: A014077 A014078 A014079 * A014081 A014082 A014083

Keyword

nonn,fini,full,base

Author

David W. Wilson

Status

approved